To the best of our knowledge, a key conceptual difference with all related works in the literature is Exact vs. Approximate Hessian: The second term on the RHS, \(J^\top \Sigma ^{-1} J\)  is identical in all the update formulations, but comes from different derivation. In the formulation of [1]} (and others) this is an approximation of the hessian, that happens to guarantee negative definiteness. In our formulation, this term comes from the linearization. This term is the exact hessian, i.e. no approximation, such that negative definiteness is implied by the linearization. This results in a more intuitive understanding of the linearization error.
